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bioNRICH is the area of the stemNRICH site devoted to the
mathematics underlying the study of the biological sciences,
designed to help develop the mathematics required to get the most
from your study of biology at A-level and university.

The Genes of Gilgamesh

Stage: 4 Challenge Level:

It is important to realise in this question that the cross is not
like a normal Mendelian cross, but that all previously accumulated
genetic infortmation is inherited.

It would be impossible to create an offspring who was two-thirds G
and one-third M. This can be explained as follows: each successive
generation doubles the number of possible 'parts' that it can be
made out of. For example the first generation is purely G or M,
whereas the second generation has two 'parts' - it can be GG, MM or
GM. Additionally, the third generation has four parts and can be
GGGG, GGGM, GGMM, GMMM or MMMM. Thus, the number of 'parts' is
clearly $2^n$ where n is an integer.

In order to be able to be composed on one-third M, we are
essentially asking if there is a value of n such that
$\frac{2^n}{3}$ is an integer. It can be seen that there is no
value of n to make this true because $2^n$ generates numbers which
are divisible only by the prime number 2, but by no other primes.
Because 3 is a prime number, this means that the expression can
never yield an integer.

We are looking to find a composition which is within 1% of
$\frac{1}{3}$. which is equivalent to the range $\frac{99}{300}$ -
$\frac{101}{300}$ which is $0.33 - 0.33\dot{6}$.

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